\[ \int \frac {1}{x^3 \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{b \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {6 \sqrt {c \,x^{4}+b \,x^{2}}}{5 b^{2} x^{6}}+\frac {8 c \sqrt {c \,x^{4}+b \,x^{2}}}{5 b^{3} x^{4}}-\frac {16 c^{2} \sqrt {c \,x^{4}+b \,x^{2}}}{5 b^{4} x^{2}} \]
command
integrate(1/x^3/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {c^{3} x}{\sqrt {c x^{2} + b} b^{4} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} c^{\frac {5}{2}} - 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} b c^{\frac {5}{2}} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{2} c^{\frac {5}{2}} - 50 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{3} c^{\frac {5}{2}} + 11 \, b^{4} c^{\frac {5}{2}}\right )}}{5 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{5} b^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]________________________________________________________________________________________